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CONTINUOUS CATEGORIES and EXPONENTIABLE TOPOSES Peter Journal of Pure and Applied Algebra 25 (1982) 255-296 2JS North-Holland Publishing Company CONTINUOUS CATEGORIES AND EXPONENTIABLE TOPOSES Peter JOHNSTONE Dept. of Pure Mathematics, University of Cambridge, Cambridge, CB2 ISB, England and Andre JOYAL D&t. de Matkmatiques. Universitk du Quebec ri Montre‘al, MontrtiaI, H3C 3P8, Canada Communicated by P.J. Freyd Received 13 October 1981 Introduction The main objective of this paper is to contribute to the study of toposes as ‘generalized spaces’, by obtaining conditions for the existence of ‘function spaces’ (i.e. exponentials) in the 2-category of (Grothendieck) toposes and geometric morphisms. In view of the importance of function spaces in many areas of topology, we feel that this objective requires little justification. However, in analysing the notion of function space in the topos-theoretic context, we have been led to introduce a new concept which we have christened a ‘continuous category’, and which seems likely to be of considerable independent interest for the light which it sheds on the rapidly growing subject of continuous lattices. Accordingly, the first two sections of the paper are devoted to developing this new concept, and it therefore seems worthwhile to preface them with a reasonably non-technical account of how it has arisen. It has been known for many years [6] that, if X and Y are spaces, the space Yx of continuous functions from X to Y has its most pleasing properties if X is locally compact. It was first pointed out by Day and Kelly [2] that this good behaviour is related to a certain lattice-theoretic property of the open-set lattices of locally compact spaces. Subsequently, lattices with this property were studied (and given the name ‘continuous lattices’) by Scott [33], for rather different reasons: his researches in the theory of computation led him to regard them as a natural generalization of algebraic lattices (in fact the completion of the latter under splitting of idempotents in a suitable category of ‘continuous maps’). More strikingly, Scott also showed that every continuous lattice admits a certain intrinsic topology, and that the spaces obtained in this way from continuous lattices are 0022-4049/82/0000-0000/$02.75 0 1982 North-Holland 256 P. Johnslone. A. Joyal precisely the injective objects (with respect to subspace inclusions) in the category of To-spaces and continuous maps. Later, work of Isbell [19], Hofmann and Lawson [15] and Banaschewski [l] emphasized the links between local compactness, exponentiability (=possessing well-behaved function-spaces), and having continuous open-set lattice. Hyland [17] took a further step in this direction when he replaced the category of spaces by that of locales (i.e. complete Heyting algebras, regarded as generalized open-set lattices [18]); he was able to show that the link between continuous lattices and exponenti- ability remained valid in this context. However, none of these authors made explicit the link between exponentiability and injectivity in the category of To-spaces (or of locales); since this link is one of the important elements in our approach to exponentiability, we sketch it here. Suppose X is a To-space (or a locale, according to taste) which is exponentiable; i.e., the functor (-) xX has a right adjoint (-)x. Let S denote the Sierpinski space, i.e. the two-point space with just one open point. It is trivial to verify that S is injective (see [33]). Moreover, the functor (-) xX preserves subspace inclusions, so its right adjoint (-)x must preserve injectives; hence Sx is injective, and by Scott’s theorem its points form a continuous lattice in a canonical way. But by the adjunction, points of Sx correspond bijectively to continuous maps X-S, and hence to open subsets of X, so these too form a continuous lattice. (Of course, it is necessary to check that the canonical ordering on points of Sx coincides with the inclusion ordering on open subsets of X, we omit the details.) In the above argument, we used only the existence of the particular exponential Sx. But this is no accident: since S is a cogenerator for the category of sober spaces, the existence of Sx implies the existence of Yx for any sober space Y. In the converse direction, suppose we know that the open sets of X form a continuous lattice. Endowing this lattice with Scott’s topology, we obtain an injective space which is clearly the only possible candidate for the exponential Sx. Once again, some further work is needed to show that this space does have the universal property of an exponential, and we shall not give the details here. Let us now compare these arguments with what happens in the category 23?op/y of Grothendieck toposes. (Here Y denotes ‘the’ classical topos of sets, but in practice it could easily be replaced by any base topos having a natural number object.) The first thing to note is that, since ‘$J’top/9’ is a 2-category, we must concern ourselves with exponentiability in the 2-categorical sense; that is, we shall say a topos A is exponentiable if there exists a functor (-)# and a natural equivalence of categories 82op/.Y( r? x, 6, .i) = 233sop/~( C?,F ) for any pair of toposes (.6 ~9) (rather than a bijection between the objects of these categories). In ‘?J2op/.r/ the role of Sierpiriski space is played by the object classifier 9 [Xl (see [24]); it is easily deduced from the universal property of this topos that it is injective Conrinuous categories and e.rponenriable roposes 25-l (with respect to subtopos inclusions) and a cogenerator in a suitable 2-categorical sense. Accordingly, we may reduce the question of whether a topos (5 is exponentiable to that of whether the particular exponential .‘/[Xl” exists; and if this exponential exists it is necessarily an injective topos. But by the adjunction, the category of points of .Y[Xl” (i.e. geometric morphisms .Y-+ .‘/[Xl”) is equivalent to C: itself. Conversely, if A is equivalent to the category of points of an injective topos, then that topos (which, as we shall see, is determined up to equivalence by its category of points) is the natural candidate for the exponential .Y[Xl’, and we shall show that it does indeed have the right universal property. Our main result on exponentiability may thus be summarized as follows: Theorem. For a bounded .‘/-topos 6, the following are equivalent: (i) A is exponentiable in B?op/.v: (ii) The exponential .Y [Xl’ exists. (iii) 6 is equivalent to the category of points of an injective topos. The proof of this theorem will occupy Section 4 of this paper. However, before we embark on its proof, it is clearly advisable to study injective toposes and their categories of points in some detail. The first investigation of injective toposes was carried out by Johnstone [21]; although this investigation was incomplete in certain important respects, it did establish the fact that the injective toposes are precisely the retracts in %32op/Y of functor categories [r!““9 91 where % has finite limits. (Note: we shall refrain from using the usual exponential notation for functor categories, since we wish to reserve it for topos exponentials.) Thus the categories of points of injective toposes are retracts, in an appropriate category, of the categories of points of such functor categories - but it is well known that the category of points of [VP, .c/‘I is equivalent to the category of flat (=left exact) covariant functors V-19’. And the categories which arise in this way are exactly the locally finitely presentable categories of Gabriel and Ulmer [7]. Thus we are led to seek a categorical characterization of the idempotent- completion of locally finitely presentable categories - which is very reminiscent of Scott’s characterization of continuous lattices as the idempotent-completion of algebraic lattices. When we have this characterization, it turns out that we can reverse the implication in the last paragraph: if C?is a retract of a locally finitely pre- sentable category, then there exists an injective topos, determined up to equivalence by d, whose category of points is equivalent to 6. But there is another direction in which we can generalize this result. It was first pointed out by Markowsky [28] that the concept of continuous lattice has a natural generalization to posets which are not necessarily lattices, and that the resulting ‘continuous posets’ have a number of useful applications. In the same way, it seems profitable to develop our ‘continuous categories’ in a context which does not require the existence of finite limits or colimits, and this is what we shall do in Section 2. We shall then be able to extend our results on injective toposes to the class of ail toposes 258 P. Johnstone. A. Jo_val which occur as retracts of presheaf toposes; the latter appears as the natural analogue of the ‘projective sober spaces’ of Hoffmann [14], i.e. the spaces obtained by endowing continuous posets with the Scott topology. What then is our definition of a continuous category? Clearly, we should seek it by taking the definition of a continuous poset and generalizing it to suit the context where the underlying structure is a category rather than a partial order. But we must exercise some care here. The usual definition of a continuous poset or lattice is phrased in terms of the properties of a certain auxiliary relation (the ‘way-below relation’) on the elements of the poset; whilst an analogue of the way-below relation (the concept of ‘wavy arrow’) certainly exists in a continuous category, it seems hard to give an intrinsic characterization of it, and it is therefore not convenient to use it in a definition.
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